Maximum point of a function about binomial coefficients

For a fixed positive integer $m$, define $$g(d)=\sum_{j=d+1}^m\binom{m}{j}\binom{j-1}{d},\ \ 1\leq d\leq m-1.$$

I used MATLAB to calculate the value of $g$ and it seems that $g(d)$ is maximal iff $d=\left[\frac{m-1}{3}\right]$.

My try: Using the formula $$\sum_{k=q}^n\binom{n}{k}\binom{k}{q}=2^{n-q}\binom{n}{q},$$ the problem is equivalent to that $$\sum_{j=0}^{m-d}\frac{d}{m-j}\binom{m-d}{j}\leq 2^{m-d-1}\Leftrightarrow d\leq\left[\frac{m-1}{3}\right].$$

I don't know what to do next.

Edit: The problem is reduced to $$\sum_{j=0}^{m-d-1}\binom{m}{j}\left(-\frac{1}{2}\right)^{m-d-j-1}\geq \binom{m}{d}\Leftrightarrow d\leq\left[\frac{m-1}{3}\right].$$